Question

A function $f:R\to R$ is defined as $f\left(x\right)=-x^3+3x^2-2x+4.$ Which of the following is true about the function $f\left(x\right)$?

• A. $f\left(x\right)$ is an onto function
• B. $f\left(x\right)$ is an into function
• C. $f\left(x\right)$ is a one one function
• D. $f\left(x\right)=4$ will have only one solution

Answer 1

The correct option is A $f\left(x\right)$ is an onto function

As $f\left(x\right)$ is a cubic polynomial,
$\therefore$ Range of $f\left(x\right)$ is $R.$
So, $f\left(x\right)$ is an onto function.

$f\left(x\right)=4\Rightarrow -x^3+3x^2-2x+4=4\Rightarrow -x(x^2-3x+2)=0\Rightarrow -x(x-1)(x-2)=0$
$\Rightarrow f\left(x\right)=4$ has three solutions.
Hence, $f\left(x\right)$ is a many-one function.